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Lecture 6 Digital Electronics – Karnaugh Maps A.D Simplifying Boolean Expressions Simplify using Boolean algebra: AB AB . AB AB Original expression A( B B ) After applying distributive law A1 After applying inverse law A After applying identity law Therefore, AB AB A . Karnaugh Maps (K –Map) To use a Karnaugh map to simplify an expression: 1. Draw a “map” with one box for each possible product term in the expression. The boxes must be arranged so that a one-box movement either horizontal or vertical changes one and only one variable. See Figure 1. 2. For each product term in the expression to be simplified, place a checkmark in the box whose labels are the product's variables and their complements. 3. Draw loops around adjacent pairs of checkmarks. Blocks are "adjacent" horizontally and vertically only, not diagonally. A block may be "in" more than one loop, and a single loop may span multiple rows, multiple columns, or both, so long as the number of checkmarks enclosed is a power of two. 4. For each loop, write an unduplicated list of the terms which appear; i.e. no matter how many times A appears, write down only one A. 5. If a term and its complement both appear in the list, e.g. both A and A , delete both from the list. 6. For each list, write the Boolean product of the remaining terms. 7. Write the Boolean sum of the products from Step 5; this is the simplified expression. Karnaugh Maps for Expressions of Two Variables Example 2: Simplify the above expression ( AB AB ) using K – Map. Solution: Draw a rectangle and divide it so that there is a row or column for each variable and its complement. Place checks in the boxes that represent each of the product terms of the expression. Draw a loop around adjacent pairs of checks. The loop contains A, B, A, and B . We remove one A so that the list is unduplicated. The B and B "cancel," leaving only A, which is the expected result: AB AB A . Example 3: simplify AB AB AB Diploma in Environmental Science/Advanced Diploma in industrial Laboratory Technology //Electronics Instrumentation – G6601/S-I/Y-09 Lecture 6 Digital Electronics – Karnaugh Maps A.D Solution Place a check in the A B area. Place a check in the AB area. Place a check in the A B area. Draw loops around pairs of adjacent checks. Because there are two loops, there will be two terms in the simplified expression. The vertical loop contains A , B, A , and B . We remove one A to make an unduplicated list. The B and B cancel, leaving the remaining A . From the horizontal loop we remove the duplicate B , then remove A and A leaving only B in the second term. We write the Boolean sum of these, and the result is A B , so: AB AB AB A B Expressions of Three Variables Recall that an essential characteristic of a Karnaugh map is that moving one position horizontally or vertically changes one and only one variable to its complement. For expression of three variables, the basic Karnaugh diagram is shown below. As with the diagram for two variables, adjacent squares differ by precisely one literal. The left and right edges are considered to be adjacent, as though the map were wrapped around to form a cylinder. Example 4 shows deriving a circuit from the truth table using the sum of products method, simplifying the sum of products expression, and drawing the new, simpler circuit. Figure 4. a) A truth table with product terms, b) the resulting sum-of-products expression (min terms), and c) the equivalent digital logic circuit. The Karnaugh map for the expression in Figure 4 b is shown below. In this Karnaugh map, the large loop surrounds ABC and ABC ; note that it "wraps around" from the left edge of the map to the right edge. The A and A cancel, so these two terms simplify to BC. Diploma in Environmental Science/Advanced Diploma in industrial Laboratory Technology //Electronics Instrumentation – G6601/S-I/Y-09 Lecture 6 Digital Electronics – Karnaugh Maps A.D A BC is in a cell all by itself, and so contributes all three of its terms to the final expression. The simplified expression is BC ABC . Simplified circuit is shown in Figure 6. Figure 6. Simplified circuit for the truth table of Figure 4a. Example 5: Generate a sum of product expression from the truth table in Figure 7a, simplify it using K - Map and draw a simpler circuit. Figure 7: a) A truth table with product terms, b) the resulting sum-of-products expression, and c) the equivalent digital logic circuit. Figure 8 shows the K-Map of the expression. Figure 8: Karnaugh map for the expression of figure 7 Diploma in Environmental Science/Advanced Diploma in industrial Laboratory Technology //Electronics Instrumentation – G6601/S-I/Y-09 Lecture 6 Digital Electronics – Karnaugh Maps A.D After removing duplicates, the large loop contains A and A and also C and C ; these cancel. All that's left after removing the two complement pairs is B. The small loop contains B and B , which are removed, so it yields AC. The simplified expression in Figure 7b is B AC Figure 9: Simplified circuit equivalent to Figure 7c. Electronic Logic Electronic logic circuits work with two levels of voltage: Low: 0 V High: The positive supply voltage. In most logic circuit, „high‟ is always 5 V. Usually the low voltage level corresponds to logical „0‟ and the high level to logical „1‟. Logic Circuits have practical applications. A simple example is shown below. Example 6 There are two switches in the circuit below. There is one lamp. The circuit has two binary inputs and one binary output. There are four possible ways in which the two switches can be set: 1. A open and B open: lamp off. 2. A closed but B open: lamp off. 3. A open but B closed: lamp off. 4. A closed AND B closed: lamp ON. The action of the circuit can be summarized by representing the binary states of inputs and output by „0‟ and „1‟. For the switches, 0 = „switch open‟ and 1 = „switch closed‟. For the lamp, 0 = „lamp off‟ and 1 = „lamp on‟. The four states of the switches can be set in a truth table. Example 7: Sprinkler Control A system is designed to turn on a garden water sprinkler when the soil is dry, but not when the sun is shining. A soil moisture sensor A has outputs 0 = moist, and 1 = dry. A light sensor B has Diploma in Environmental Science/Advanced Diploma in industrial Laboratory Technology //Electronics Instrumentation – G6601/S-I/Y-09 Lecture 6 Digital Electronics – Karnaugh Maps A.D outputs 0 = dull, and 1 = sunny. For the sprinkler S, 0 = off, and 1 = on. The truth table for the system is shown below. Draw the logic circuit for the control. Inputs Outputs A B S 0 0 0 0 1 0 1 0 1 1 1 0 Solution: The third line down corresponds to „dry soil and dull day‟, the ideal conditions for watering. For S = 1, we see that A = 1 and B = 0. As a logic equation, this is written: S AB Note: Building logic circuits is simple. All the logic gates and other more complicated logic circuits that you might need are available as integrated circuits (ICs). There are two commonly used „families‟ of logic ICs: TTL - short for transistor-transistor logic. This runs on 5 V, so it needs a regulated power supply. TTL type numbers all begin with „74‟ so this is sometimes known as the 74XX series. There are various types of TTL, of which the Low Power Schottky type has almost replaced the original 74XX series. 74LSXX ICs need less power than the 74XX type. CMOS - complementary Metal Oxide Semiconductor. These types have numbers ranging upward from 4000, so are sometimes known as the „4000‟ series. There is also the „4500‟ series. Members of both series run on any voltage between 3 V and 15 V. CMOS is slower than TTL but requires less current. It has the additional advantage that it does not require a regulated power supply. Both TTL and CMOS are packaged as double in- line ICs. They usually have 14 or 16 pins. There are four AND gates to the 7408, 74LS08, and 4081 ICs. The four gates share the power supply pins. TTL inputs may be left unconnected. An unconnected input behaves as if it has a high (=‟1‟) signal applied to it. Diploma in Environmental Science/Advanced Diploma in industrial Laboratory Technology //Electronics Instrumentation – G6601/S-I/Y-09 Lecture 6 Digital Electronics – Karnaugh Maps A.D Tutorial 6 1. Use a Karnaugh map to simplify ABC ABC ABC 2. In a certain chemical-processing plant, a liquid chemical is used in a manufacturing process. The chemical is stored in three different tanks. A level sensor in each tank produces a HIGH voltage when the level of chemical in the tank drops below a specified point. Design a circuit that monitors the chemical level in each tank and indicates when the level in any two tanks drops below the specified point. (hint: use Karnaugh Map to Simplify expression) 3. Simplify the following circuit. b) c) a) 4. State the properties of TTL and CMOS logic gates. Diploma in Environmental Science/Advanced Diploma in industrial Laboratory Technology //Electronics Instrumentation – G6601/S-I/Y-09

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